YES(O(1),O(n^2)) 184.98/60.02 YES(O(1),O(n^2)) 184.98/60.02 184.98/60.02 We are left with following problem, upon which TcT provides the 184.98/60.02 certificate YES(O(1),O(n^2)). 184.98/60.02 184.98/60.02 Strict Trs: 184.98/60.02 { +(0(), y) -> y 184.98/60.02 , +(s(x), 0()) -> s(x) 184.98/60.02 , +(s(x), s(y)) -> s(+(s(x), +(y, 0()))) } 184.98/60.02 Obligation: 184.98/60.02 derivational complexity 184.98/60.02 Answer: 184.98/60.02 YES(O(1),O(n^2)) 184.98/60.02 184.98/60.02 The weightgap principle applies (using the following nonconstant 184.98/60.02 growth matrix-interpretation) 184.98/60.02 184.98/60.02 TcT has computed the following triangular matrix interpretation. 184.98/60.02 Note that the diagonal of the component-wise maxima of 184.98/60.02 interpretation-entries contains no more than 1 non-zero entries. 184.98/60.02 184.98/60.02 [+](x1, x2) = [1] x1 + [1] x2 + [1] 184.98/60.02 184.98/60.02 [0] = [0] 184.98/60.02 184.98/60.02 [s](x1) = [1] x1 + [0] 184.98/60.02 184.98/60.02 The order satisfies the following ordering constraints: 184.98/60.02 184.98/60.02 [+(0(), y)] = [1] y + [1] 184.98/60.02 > [1] y + [0] 184.98/60.02 = [y] 184.98/60.02 184.98/60.02 [+(s(x), 0())] = [1] x + [1] 184.98/60.02 > [1] x + [0] 184.98/60.02 = [s(x)] 184.98/60.02 184.98/60.02 [+(s(x), s(y))] = [1] y + [1] x + [1] 184.98/60.02 ? [1] y + [1] x + [2] 184.98/60.02 = [s(+(s(x), +(y, 0())))] 184.98/60.02 184.98/60.02 184.98/60.02 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 184.98/60.02 184.98/60.02 We are left with following problem, upon which TcT provides the 184.98/60.02 certificate YES(O(1),O(n^2)). 184.98/60.02 184.98/60.02 Strict Trs: { +(s(x), s(y)) -> s(+(s(x), +(y, 0()))) } 184.98/60.02 Weak Trs: 184.98/60.02 { +(0(), y) -> y 184.98/60.02 , +(s(x), 0()) -> s(x) } 184.98/60.02 Obligation: 184.98/60.02 derivational complexity 184.98/60.02 Answer: 184.98/60.02 YES(O(1),O(n^2)) 184.98/60.02 184.98/60.02 We use the processor 'matrix interpretation of dimension 2' to 184.98/60.02 orient following rules strictly. 184.98/60.02 184.98/60.02 Trs: { +(s(x), s(y)) -> s(+(s(x), +(y, 0()))) } 184.98/60.02 184.98/60.02 The induced complexity on above rules (modulo remaining rules) is 184.98/60.02 YES(?,O(n^2)) . These rules are moved into the corresponding weak 184.98/60.02 component(s). 184.98/60.02 184.98/60.02 Sub-proof: 184.98/60.02 ---------- 184.98/60.02 TcT has computed the following triangular matrix interpretation. 184.98/60.02 184.98/60.02 [+](x1, x2) = [1 0] x1 + [1 1] x2 + [0] 184.98/60.02 [0 1] [0 1] [0] 184.98/60.02 184.98/60.02 [0] = [0] 184.98/60.02 [0] 184.98/60.02 184.98/60.02 [s](x1) = [1 0] x1 + [0] 184.98/60.02 [0 1] [1] 184.98/60.02 184.98/60.02 The order satisfies the following ordering constraints: 184.98/60.02 184.98/60.02 [+(0(), y)] = [1 1] y + [0] 184.98/60.02 [0 1] [0] 184.98/60.02 >= [1 0] y + [0] 184.98/60.02 [0 1] [0] 184.98/60.02 = [y] 184.98/60.02 184.98/60.02 [+(s(x), 0())] = [1 0] x + [0] 184.98/60.02 [0 1] [1] 184.98/60.02 >= [1 0] x + [0] 184.98/60.02 [0 1] [1] 184.98/60.02 = [s(x)] 184.98/60.02 184.98/60.02 [+(s(x), s(y))] = [1 1] y + [1 0] x + [1] 184.98/60.02 [0 1] [0 1] [2] 184.98/60.02 > [1 1] y + [1 0] x + [0] 184.98/60.02 [0 1] [0 1] [2] 184.98/60.02 = [s(+(s(x), +(y, 0())))] 184.98/60.02 184.98/60.02 184.98/60.02 We return to the main proof. 184.98/60.02 184.98/60.02 We are left with following problem, upon which TcT provides the 184.98/60.02 certificate YES(O(1),O(1)). 184.98/60.02 184.98/60.02 Weak Trs: 184.98/60.02 { +(0(), y) -> y 184.98/60.02 , +(s(x), 0()) -> s(x) 184.98/60.02 , +(s(x), s(y)) -> s(+(s(x), +(y, 0()))) } 184.98/60.02 Obligation: 184.98/60.02 derivational complexity 184.98/60.02 Answer: 184.98/60.02 YES(O(1),O(1)) 184.98/60.02 184.98/60.02 Empty rules are trivially bounded 184.98/60.02 184.98/60.02 Hurray, we answered YES(O(1),O(n^2)) 184.98/60.04 EOF